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Meandering of trajectories of polynomial vector fields in the affine n-space.

Dimitri Novikov, Sergei Yakovenko (1997)

Publicacions Matemàtiques

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in Rn and an affine hyperplane.The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives.This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted...

Modular deformations and space curve singularities.

Bernd Martin (2003)

Revista Matemática Iberoamericana

We investigate different concepts of modular deformations of germs of isolated singularities (infinitesimal, Artinian, formal). An obstruction calculus based on the graded Lie algebra structure of the tangent cohomology for modular dcformations is introduced. As the main result the characterisation of the maximal infinitesimally modular subgerm of the miniversal family as flattening stratum of the relative Tjurina module is extended from ICIS to space curve singularities.

Multidimensional term indexing for efficient processing of complex queries

Michal Krátký, Tomáš Skopal, Václav Snášel (2004)

Kybernetika

The area of Information Retrieval deals with problems of storage and retrieval within a huge collection of text documents. In IR models, the semantics of a document is usually characterized using a set of terms. A common need to various IR models is an efficient term retrieval provided via a term index. Existing approaches of term indexing, e. g. the inverted list, support efficiently only simple queries asking for a term occurrence. In practice, we would like to exploit some more sophisticated...

Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.

Francisco Santos (1997)

Revista Matemática de la Universidad Complutense de Madrid

We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance. Our method works if the topological model T to be realized has only double singularities and gives an algebraic curve of degree 2N+2K, where N and K are the numbers of double points and connected components of T. This degree is optimal in the sense that for any choice of the numbers N and K there exist models which cannot be realized algebraically with...

Polynomial bounds for the oscillation of solutions of Fuchsian systems

Gal Binyamini, Sergei Yakovenko (2009)

Annales de l’institut Fourier

We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension n having m singular points. As a function of n , m , this bound turns out to be double exponential in the precise sense explained in the paper.As a corollary, we obtain a solution of the so-called restricted infinitesimal...

Puiseux Expansion of a Cuspidal Singularity

Maciej Borodzik (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We present an effective and elementary method of determining the topological type of a cuspidal plane curve singularity with given local parametrization.

Quasi-homogeneous linear systems on ℙ² with base points of multiplicity 7, 8, 9, 10

Marcin Dumnicki (2011)

Annales Polonici Mathematici

We prove that the Segre-Gimigliano-Harbourne-Hirschowitz conjecture holds for quasi-homogeneous linear systems on ℙ² for m = 7, 8, 9, 10, i.e. systems of curves of a given degree passing through points in general position with multiplicities at least m,...,m,m₀, where m = 7, 8, 9, 10, m₀ is arbitrary.

Rational Bézier curves with infinitely many integral points

Petroula Dospra (2023)

Archivum Mathematicum

In this paper we consider rational Bézier curves with control points having rational coordinates and rational weights, and we give necessary and sufficient conditions for such a curve to have infinitely many points with integer coefficients. Furthermore, we give algorithms for the construction of these curves and the computation of theirs points with integer coefficients.

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